NOTES

1. For example, at the end of the definitions of Book I:,"I use the words 'attraction', 'impulse', or 'tendency' of anything toward the center interchangeably with one another, considering these forces not physically but only mathematically. Whence the reader should beware lest he think that by such words I am anywhere defining a species or mode of action or a physical cause or explanation, or that I am actually and physically attributing forces to centers (which are mathematical points), if perhaps I should say either that centers attract or that centers have forces." Newton, 1687, pp. 4-5.

2. So it appeared to Joseph Needham viewing the origins of modern science from the perspective of his study of Chinese science; see Needham 1959, Sect. 19(k).

3. "Sed μηχανικην eam Geometriae partem intelligimus, quae Motum tractat: atque Geometricis rationibus, & αποδεικτικως, inquirit, quâ vi quisque motus peragatur." Wallis 1670, p. 1. Cf. Newton 1687, Praef. ad lect.: "Quo sensu Mechanica rationalis erit Scientia Motuum qui ex viribus quibuscunque resultant, & virium quae ad motus quoscunque requiruntur, accurate proposita ac demonstrata." On the varied and changing meanings of "mechanics" in the 16th and 17th centuries, see Gabbey 1992.

4. Galileo 1969, Vol. I, p. 281.

5. Fontenelle 1725, p. 87.

6. Newton 1687, Book I, Sect. VI, Lemma XXVIII: "There is no oval figure, of which the area cut off by any straight line can be found generally by means of equations finite in the number of terms and dimensions."

7. See Leibniz 1686 and Leibniz 1694a. Cf. Bos 1988.

8. Vlastos 1975.

9. Hatfield 1990.

10. "It seems to me that the everyday practice of the famous Arsenal of you gentlemen of Venice, in particular in that part called 'mechanics', opens to speculative minds a wide field for philosophizing." Galileo 1638, p. 1. Cf. Robert Boyle's (1627-91) brief argument "That the Goods of Mankind may be much Increased by the Naturalist's Insight into Trades", resting on the propositions that "the phaenomena afforded by the trades are (most of them) a part of the history of nature, and therefore may both challenge the naturalist's curiosity, and to [!] add to his knowledge," and that "[the phaenomena of trades] show us nature in motion, and that too, when she is (as it were) put out of her course, by the strength or skill of man, which I have formerly noted to be the most instructive condition, wherein we can behold her." (Works, III, 442-3; as published in Boyle 1965, pp. 163-5).

11. Kepler to Herwart von Hohenberg, 10 Feb. 1605, in Kepler 1937--, vol.. XV, p. 146.

12. Gabbey 1990. For a more extensive survey of the changing meaning of "mechanics" in the seventeenth century, see Gabbey 1992.

13. For a survey and analysis of various notions of perpetual motion and their relation to the principles of mechanics in the seventeenth century, see Gabbey 1985.

14. Stevin 1586, Book I, Prop. 19, pp. 175-179. One may question the cogency of Stevins' reasoning here, to wit, whether the perpetual motion he describes is in fact impossible in the abstract (see Gabbey 1985, 74, n.26). But the point at hand concerns his translation of its practical impossibility into a mathematical argument.

15. The frontispiece of Fontana's treatise on moving the obelisk graphically illustrates the problem of scaling, as many of the competing proposals portrayed there are clearly unfeasible when scaled to the full-sized obelisk.

16. Letter to Castelli, 21.XII.1613; in Galileo 1969, vol. I, p. 177.

17. See, for example, Chap. [14] of his De motu [antiquiora] (ca. 1590) in Galileo 1960, p.64.

18. On Galileo's early work in mechanics, see Clavelin 1974, Chap. 3.

19. On the relation of Galileo's new science of motion to medieval sources as mediated by the sixteenth-century scholastic curriculum, see Wallace 1984, esp. Chap. 4.

20. On the "mean speed theorem" and medieval kinematics in general, see Clagett 1959, esp. Part II.

21. For an overview, see Clagett 1968.

22. Molland 1982.

23. Galileo 1638, p. 28.

24. Cavalieri to Galileo, 2 October 1634, in Galileo 1890-1909, vol. XVI, pp. 136-7; 19 December 1634, Ibid., pp. 175-6.

25. On the general question of the concept of momentum in Galileo's thought, see Galluzzi 1979.

26. Schuster 1977, pp. 299-352.

27. A cycloid is the path traced by a fixed point on the circumference of a circle as the circle rolls along a straight line; see below, §4.2.

28. For a general account of Huygens' mechanics see Gabbey 1980; for its role in his work on clocks and the determination of longitude, see Mahoney 1980.

29. Galileo had first called attention to the pendulum as a system that would continue to oscillate uniformly, falling and then rising to the same height, were it not for external damping forces. He established that the period is independent of the weight of the bob and asserted that it is also independent of the amplitude. Although Galileo saw the pendulum as a time-keeping device and suggested attaching it to the escapement of a clock, he himself used it primarily as a means of experimenting with falling bodies, using it to argue the speed of fall is independent of weight and that a body in falling acquires enough momento or impeto to raise it to its original height. See Bedini 1991.

30. Torricelli, De motu gravium naturaliter descendentium et projectorum, in Torricelli 1919, vol. 2, p.105. E.J. Dijksterhuis first brought out the relation between Torricelli and Huygens; see Dijksterhuis 1961, pp. 370-2.

31. Huygens 1659, Prop.XI.

32. Note, for example, the similarity between Huygens' remark in 1675 that "the quantity of incitation at each instant of motion is measured by the force required to prevent the body from starting to move at the place where it is and in the direction it is headed," (Huygens 1888, Vol. XVIII, p. 496) and Newton's definition of an impressed force as "the action carried out on a body to change its state either of motion or of moving uniformly in a straight line" (Newton 1687, Book I, def. 4). Cf. Gabbey 1980, pp. 176-77.

33. Herivel 1965, 129-130, from Newton's Waste Book in a section dating from 1664. Newton recalled this earlier derivation in a scholium to Proposition I,4 of the Principia (Newton 1687), where he emphasized the total force exerted by successive impacts. Over a given period of time, he reasoned, that force is proportional to the velocity and to the number of reflections. For polygon of any given number of sides, the velocity will be as the distance traveled, and the number of reflections will be as that distance divided by the length of a side, which in turn is proportional to the radius. Hence the total force will be as the square of the distance divided by the radius, "and thus, if the polygon with infinitely diminished sides coincides with the circle, as the square of the arc described in the given time divided by the radius." Again the transition from inscribed polygons to the circle as limit rests on a property that is seemingly independent of the number or size of the sides.

34. For greater detail and further references, see Mahoney 1993.

35. Newton 1730, p. 396.

36. See Heilbron 1993.

37. Descartes to Mersenne, 28.VIII.1638, in Descartes 1879-1913, II, 309.

38. Ibid., 313: "One should also note that curves described by rolling circles (roulettes) are entirely mechanical lines and count among those I have rejected from my Geometry; that is why it is no wonder that their tangents are not found by the rules I have set out there."

39. Roberval 1693, pp. 1-89.

40. "To resolve problems by motion", <October, 1666>, in Newton 1967, vol. I, pp. 400-48.

41. As a simple example, let y2 = x3. Adding moments, (y + qo)2 = (x + po)3, or y2 + 2yqo + q2o2 = x3 + 3x2po + 3xp2o2 + p3o3. Deleting the equal terms and dividing the others by o yields 2yq + q2o = 3x2p + 3xp2o + p3o2. Since o is infinitely small, so too are all terms containing it, whence 2yq = 3x2p, or y2(2q/y) = x3(3p/x), as the rule states.

42. Newton, "De analysi per aequationes numero terminorum infinitas", in Newton 1967, vol. II, pp. 206-47; cf. the fully developed (but long unpublished) tract "On the Methods of Series and Fluxions" (1671), in Newton 1967, vol. III, pp. 32-353. Despite claims made during the priority dispute with Leibniz, Newton did not replace p and q with the now familiar dot notation, or "pricked letters" and , until 1691 (Ibid., 72, n. 86).

43. Leibniz 1855, vol. I, app. II; summarized in English with translated excerpts in Child 1920, 65ff.

44. For a discussion of the Aristotelean passages (Metaphysics VI, 1,1026a23-7, and XI, 7, 1064b8-9) and their subsequent interpretation, see Sasaki 1988, Chap.6. More generally, see Crapulli 1965.

45. In artem analyticen isagoge, in Viète 1646, p. 1: Est veritatis inquirendae via quaedam in Mathematicis, quam Plato primus invenisse dicitur, à Theone nominata Analysis, & ab eodem definita, Adsumptio quaesiti tanquam concessiper consequentia ad verum concessum. Ut contrà Synthesis, Adsumptio concessi per consequentia ad quaesiti finem & comprehensionem.

46. The precise meaning of Pappus' description of analysis and synthesis is a matter of dispute. See Mahoney 1968, Hintikka and Remes 1974, and, most generally, Knorr 1993.

47. Mahoney 1994, Chap. 2, and Morse 1981.

48. The constituents of his Opus restitutae mathematicae analyseos, seu algebra nova, all composed in the 1570s and '80s, were published separately over the course of several decades and brought together for the first time in Viète 1646. They include In artem analyticem isagoge (Tours, 1591), Ad logisticen speciosam notae priores (Paris, 1631), Zeteticorum libri quinque (Tours, 1593), De aequationum recognitione et emendatione tractatus duo (Paris, 1615), De numerosa potestatum ad exegesin resolutione (Paris, 1600), Effectionum geometricarum canonica recensio (Tours, 1592), Supplementum geometriae (Tours, 1593), Theoremata ad sectiones angulares (Paris, 1615), and Variorum de rebus mathematicis responsorum liber VIII (Tours, 1593). For the complex history of Viète's works, see Van Egmond 1985, and, more extensively, Grisard 1976.

49. To these, Tartaglia, Cardano, and Ferrari had added general solutions of the cubic and quartic equation. On the Arabic background and its influence on European developments, see Rashed 1984.

50. See, for example, Rule IV of the Regulae ad directionem ingenii, in Descartes 1879-1913, vol. X, p. 377: "Finally, there have been some most ingenious men who have tried in this century to revive the same [true mathematics]; for it seems to be nothing other than that art which they call by the barbarous name of 'algebra', if only it could be disentangled from the multiple numbers and inexplicable figures that overwhelm it, so that it would no longer lack the clarity and simplicity that we suppose should obtain in a true mathematics."

51. In the sense that, given any two terms of the progression, the smaller can be added to itself a sufficient number of times to exceed the greater. Adding a line to itself however many times will not produce an area. See his Géométrie (Leiden, 1637, as part of the Discours de la méthode and Essais; repr. separately with English trans. in Descartes 1954), 297-8.

52. Descartes 1954, Book III, p.380: "Au reste tant les vrayes racines que les fausses ne sont pas tousiours relles; mais quelquefois seulement imaginaires; c'est à dire qu'on peut bien tousiours en imaginer autant que iay dit en chasque Equation; mais qu'il n'y a quelquefois aucune quantité, qui corresponde a celles qu'on imagine. comme encore qu'on en puisse imaginer trois en celle cy, x3 - 6xx + 13x - 10 = 0, il n'y en a toutefois qu'une reelle, qui est 2, & pour les deux autres, quoy qu'on les augmente, ou diminue, ou multipie en la façon que ie viens d'expliquer, on ne sçauroit les rendres autres qu'imaginaires." Just over a century later, Euler commented on the analytic value of imaginaries: "We must finally drop our concern that the doctrine of impossible numbers might be viewed as useless fantasy. This concern is unfounded. The doctrine of impossible numbers is in fact of the greatest importance, since problems often arise in which one cannot know immediately whether they demand something possible or impossible. Whenever their solution leads to such impossible numbers, one has a sure sign that the problem demands something impossible." Euler 1959 [1770], Pt. I, sec. 1, par. 151.

53. Descartes 1954, Book II, p. 342.

54. For a detailed study of the development of Fermat's mathematics, see Mahoney 1994. With the exception of one treatise published anonymously in 1660, Fermat's works circulated only in manuscript copies during his lifetime. His son, Samuel, gathered a collection of them, which he issued as Fermat 1679. The main source today is Fermat 1891-1922.

55. It is, of course, one of the elementary symmetric functions of the equation. Viète's theory of equations brought out some, but not all, of these relations. Descartes focused on them as the links between an equation written as a polynomial and as a product of the linear binomials containing its roots.

56. Fermat first announced his method in an essay, "Methodus ad disquirendam maximam et minimam et de tangentibus linearum curvarum", sent to Mersenne in 1636 but based on results achieved in 1629. The essay gave no hint of its origins, which Fermat revealed only in the course of a dispute with Descartes in 1638. For details, see Mahoney 1994, Chap. 4.

57. Viewing the origins of the calculus through Bishop Berkeley's lenses obscures the lines of thought that avoided infinites and infinitesimals altogether or that sought to keep them in a separate domain from finite quantities, linked only by common relationships.

58. Grosholz 1991 offers a sustained critique of this approach both to mathematics and to physics.

59. Cavalieri 1653; see Andersen 1985.

60. Fermat refined his methods of quadrature in the late 1630s and early '40s, making them known largely through his correspondence with Roberval; see Mahoney 1994, Chap. 5. Roberval's methods similarly were known only to his network of correspondents and perhaps also to his students at the Collège royal de France (College de France after the Revolution) until the publication of his Traité des indivisibles in Roberval 1693. For Roberval's correspondence with Torricelli in the mid-1640s, on which the discussion to follow is based, see Torricelli 1919, vol. III, passim.

61. This is how the language of Cavalieri's method of indivisibles became attached to a method of infinitesimals conceptually different from it, thus creating the historical misunderstanding that the techniques of quadrature from which the integral calculus emerged stemmed from Cavalieri.

62. Fermat, De aequationum localium transmutatione et emendatione ad multimodam curvilineorum inter se vel cum rectilineis comparationem ..., [late 1650s], Fermat 1891-1922, vol. I, pp. 255-85; see Mahoney 1994, Chap. 5, Sect. IV.

63. Barrow 1670; Gregory 1668; Mahoney 1990.

64. On Barrow, see Mahoney 1990.

65. Leibniz 1684; repub. in Leibniz 1849-63, vol. V, pp. 220-6.

66. Leibniz 1684, p. 223.

67. Leibniz 1686; Leibniz 1849-63, vol. V, pp. 226-33.

68. Leibniz to Huygens, 21 September 1691, Huygens 1888, Vol. X, p. 157.

69. Leibniz to Varignon, 2 February 1702, Leibniz 1849-63, vol. IV, p. 92. "To tell the truth," he added a few months later (ibid., 110), "I am not altogether persuaded myself that our infinites and infinitesimals should be considered other than as ideal objects or as well founded fictions." Cf. Leibniz to Johann Bernoulli, 29 July 1698, Leibniz 1849-63, Vol. III, p. 524: Inter nos autem haec addo, quod et jam olim in dicto Tractatu inedito adscripsi, dubitari posse an lineae rectae infinitae longitudine et tamen terminatae revera dentur. Interim sufficere pro Calculo, ut fingantur, uti imaginariae radices in Algebra.

70. See Bos 1974.

71. Leibniz to Varignon, 2 February 1702, Leibniz 1849-63, Vol. IV, pp. 93-4.

72. Newton 1687, p. 35. Cf. de Gandt 1986.

73. On the first strategy, see Jesseph 1989.

74. Cf. Breger 1990.

75. Bacon 1960 [1620], Book I, Aph. CXXIV, p. 114; where, however, ipsissimae res is rendered "the very same things".

76. Fontenelle 1727, Préface, [bivr].

77. Barrow 1670, pp. 167-8.

78. Bacon, Nov. Org. I, Aph. 51.

79. Descartes, Le monde, Chap. 7, p.75. The final clause is quoted from the Bible, Sapientia (Wisdom), VIII, 21.

80. The discussion in the Two New Sciences of the paradoxes of the infinite may have made them understandable to a wider, popular audience, but it did not go beyond what Aristotle had said.

81. Even if, to preserve the continuity of space identified with matter, Descartes was not prepared to accept any minimal size of the particles of matter but rather posited potentially infinite subdivision; see, for example, Principles of Philosophy, II, 34.

82. Varignon takes dt as constant, i.e. ddt = 0. Doing so in Leibnizian calculus is equivalent in the modern form to choosing t as the independent variable, i.e. as the variable with respect to which one is differentiating. The Leibnizian version allows a greater flexibility in analyzing differential relations, as Bernoulli had shown in his lectures on the method of integrals and as Varignon emphasized in his later memoirs. On the mathematical point, cf. Bos 1974.

83. Leibniz 1694b, Leibniz 1694a.